A Study to Investigate Blinatumomab in Combination With Chemotherapy in Patients With Newly Diagnosed B-Lymphoblastic Leukemia

Inclusion Criteria: * All B-ALL patients must be enrolled on APEC14B1 and consented to Eligibility Screening (Part A) prior to treatment and enrollment on AALL1731. APEC 14B1 is not a requirement for B-LLy patients. B-LLy patients may directly enroll on AALL1731. * Age at diagnosis: * Patients must be \>= 365 days and \< 10 years of age (B-ALL patients without DS). * Patients must be \>= 365 days and =\< 31 years of age (B-ALL patients with DS). * Patients must be \>= 365 days and =\< 31 years of age (B-LLy patients with or without DS). * B-ALL patients without DS must have an initial white blood cell count \< 50,000/uL (performed within 7 days prior to enrollment). * B-ALL patients with DS are eligible regardless of the presenting white blood cell count (WBC) (performed within 7 days prior to enrollment). * Patient has newly diagnosed B-cell ALL, with or without Down syndrome: \> 25% blasts on a bone marrow (BM) aspirate; * OR if a BM aspirate is not obtained or is not diagnostic of B-ALL, the diagnosis can be established by a pathologic diagnosis of B-ALL on a BM biopsy; * OR a complete blood count (CBC) documenting the presence of at least 1,000/uL circulating leukemic cells; * OR patient has newly diagnosed B-cell LLy Murphy stages I or II, with or without Down syndrome. * Note: For B-LLy patients with tissue available for flow cytometry, the criterion for diagnosis should be analogous to B-ALL. For tissue processed by other means (i.e., paraffin blocks), the methodology and criteria for immunophenotypic analysis to establish the diagnosis of B-LLy defined by the submitting institution will be accepted (diagnostic biopsy for B-LLy must be performed within 14 days prior to enrollment). * All patients and/or their parents or legal guardians must sign a written informed consent. * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met. Exclusion Criteria: * Patient must not have secondary ALL that developed after treatment of a prior malignancy with cytotoxic chemotherapy. Note: patients with Down syndrome with a prior history of transient myeloproliferative disease (TMD) are not considered to have had a prior malignancy. They would therefore be eligible whether or not the TMD was treated with cytarabine. * With the exception of steroid pretreatment or the administration of intrathecal cytarabine, patients must not have received any prior cytotoxic chemotherapy for either the current diagnosis of B ALL or B LLy or for any cancer diagnosed prior to initiation of protocol therapy on AALL1731. * For patients receiving steroid pretreatment, the following additional exclusion criteria apply: * Non-DS B-ALL patients must not have received steroids for more than 24 hours in the 2 weeks prior to diagnosis without a CBC obtained within 3 days prior to initiation of the steroids. * DS and non-DS B-LLy patients must not have received \> 48 hours of oral or IV steroids within 4 weeks of diagnosis. * Patients who have received \> 72 hours of hydroxyurea within 1 week (7 days) prior to the start of systemic protocol therapy. * B-ALL patients who do not have sufficient diagnostic bone marrow submitted for APEC14B1 diagnostic testing and who do not have a peripheral blood sample submitted containing \> 1,000/uL circulating leukemia cells. * Patient must not have acute undifferentiated leukemia (AUL). * Non-DS B-ALL patients with central nervous system \[CNS\]3 leukemia (CNS status must be known prior to enrollment). * Note: DS patients with CNS3 disease are eligible but will be assigned to the DS-High B-ALL arm. CNS status must be determined based on a sample obtained prior to administration of any systemic or intrathecal chemotherapy, except for steroid pretreatment. * Non-DS B-ALL patients with testicular leukemia. (Note: DS patients with testicular disease are eligible but will be assigned to the DS-High B-ALL arm). * For LLy patients, the following additional exclusion criteria apply: * T-Lymphoblastic Lymphoma. * Morphologically unclassifiable lymphoma. * Absence of both B-cell and T-cell phenotype markers in a case submitted as lymphoblastic lymphoma. * CNS positive disease or testicular involvement. * M2 (5% - 25% blasts) or M3 (\> 25% blasts) marrow. * Patients with known Charcot-Marie-Tooth disease. * Patients with known MYC translocation associated with mature (Burkitt) B-cell ALL, regardless of blast immunophenotype. * Patients requiring radiation at diagnosis. * Female patients who are pregnant since fetal toxicities and teratogenic effects have been noted for several of the study drugs. A pregnancy test is required for female patients of childbearing potential. * Lactating females who plan to breastfeed their infants. * Sexually active patients of reproductive potential who have not agreed to use an effective contraceptive method for the duration of their study participation.